Solve for $x$ : $ 3|x + 6| - 10 = -2|x + 6| + 3 $
Answer: Add $ {2|x + 6|} $ to both sides: $ \begin{eqnarray} 3|x + 6| - 10 &=& -2|x + 6| + 3 \\ \\ { + 2|x + 6|} && { + 2|x + 6|} \\ \\ 5|x + 6| - 10 &=& 3 \end{eqnarray} $ Add ${10}$ to both sides: $ \begin{eqnarray} 5|x + 6| - 10 &=& 3 \\ \\ { + 10} &=& { + 10} \\ \\ 5|x + 6| &=& 13 \end{eqnarray} $ Divide both sides by ${5}$ $ \dfrac{5|x + 6|} {{5}} = \dfrac{13} {{5}} $ Simplify: $ |x + 6| = \dfrac{13}{5}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 6 = -\dfrac{13}{5} $ or $ x + 6 = \dfrac{13}{5} $ Solve for the solution where $x + 6$ is negative: $ x + 6 = -\dfrac{13}{5} $ Subtract ${6}$ from both sides: $ \begin{eqnarray} x + 6 &=& -\dfrac{13}{5} \\ \\ {- 6} && {- 6} \\ \\ x &=& -\dfrac{13}{5} - 6 \end{eqnarray} $ Change the ${ - 6}$ to an equivalent fraction with a denominator of $5$ $ x = - \dfrac{13}{5} {- \dfrac{30}{5}} $ $ x = -\dfrac{43}{5} $ Then calculate the solution where $x + 6$ is positive: $ x + 6 = \dfrac{13}{5} $ Subtract ${6}$ from both sides: $ \begin{eqnarray} x + 6 &=& \dfrac{13}{5} \\ \\ {- 6} && {- 6} \\ \\ x &=& \dfrac{13}{5} - 6 \end{eqnarray} $ Change the ${ - 6}$ to an equivalent fraction with a denominator of $5$ $ x = \dfrac{13}{5} {- \dfrac{30}{5}} $ $ x = -\dfrac{17}{5} $ Thus, the correct answer is $x = -\dfrac{43}{5} $ or $x = -\dfrac{17}{5} $.